The conformal model of Euclidean space is a powerful tool that embeds our familiar 3D world into a higher-dimensional space. By mapping points to null vectors and using geometric algebra, it simplifies many geometric computations and unifies the treatment of parallel and intersecting objects.
This model preserves angles and circles, making it ideal for studying Euclidean geometry. It also allows for easy representation of lines, planes, and spheres, while providing a natural way to handle the point at infinity, which is crucial for understanding geometric relationships and transformations.
Construction using the Minkowski Plane
- Embed Euclidean space into a higher-dimensional space called the Minkowski plane, a 2D vector space with a metric of signature (1,1)
- Map points in Euclidean space to null vectors in the Minkowski plane, which have a zero inner product with themselves under the Minkowski metric
- Origin of Euclidean space maps to the null vector $(1, 0, 0)$
- Point at infinity maps to the null vector $(0, 0, 1)$
- Represent lines and planes in Euclidean space as bivectors in the Minkowski plane, which are 2D subspaces spanned by two null vectors
- Conformal model preserves angles and circles from Euclidean space, making it a conformal embedding useful for studying Euclidean geometry
Properties and Advantages
- Allows for a unified treatment of parallel and intersecting lines and planes through the point at infinity
- Lines in opposite directions in Euclidean space intersect at the point at infinity $(0, 0, 1)$ in the conformal model
- Planes with opposite orientations in Euclidean space intersect at the point at infinity in the conformal model
- Represents non-Euclidean geometries (spherical and hyperbolic) by modifying the metric of the Minkowski plane
- Geometric algebra operations (inner and outer products) have natural interpretations corresponding to geometric relationships and transformations in Euclidean space
- Provides a compact and efficient representation of Euclidean geometry, simplifying many geometric computations and proofs compared to traditional vector algebra methods
Geometric Meaning of Vectors
Interpretation based on Grade
- Grade 1 vectors represent points in Euclidean space, with basis vector coefficients determining point coordinates
- Example: $(1, 2, 3, 4, 29)$ represents the point $(2, 3, 4)$ in Euclidean space
- Grade 2 bivectors represent lines and planes in Euclidean space, formed by the outer product of two null vectors representing defining points
- Example: $e_1 \wedge e_2$ represents the plane spanned by the $x$ and $y$ axes
- Grade 3 trivectors represent spheres and circles in Euclidean space, formed by the outer product of three null vectors representing points on the object
- Example: $e_1 \wedge e_2 \wedge e_3$ represents the unit sphere centered at the origin
Geometric Relationships from Inner Products
- Inner product of vectors determines geometric relationships
- Distance between points
- Angle between lines or planes
- Example: $a \cdot b = |a||b|\cos(\theta)$, where $\theta$ is the angle between lines/planes represented by bivectors $a$ and $b$
- Intersection of spheres or circles
- Example: $(e_1 \wedge e_2 \wedge e_3) \cdot (e_1 \wedge e_2 \wedge e_4)$ represents the intersection of two spheres
- Map points $(x, y, z)$ to null vectors $(1, x, y, z, x^2 + y^2 + z^2)$, where the last coordinate is the squared magnitude of the Euclidean vector
- Map lines defined by two points to bivectors by taking the outer product of the null vectors representing the points
- Example: Line through $(1, 2, 3)$ and $(4, 5, 6)$ maps to $(1, 1, 2, 3, 14) \wedge (1, 4, 5, 6, 77)$
- Map planes defined by three points to bivectors by taking the outer product of the null vectors representing any two of the three points
- Example: Plane through $(0, 0, 0)$, $(1, 0, 0)$, and $(0, 1, 0)$ maps to $(1, 0, 0, 0, 0) \wedge (1, 1, 0, 0, 1)$
- Extract Euclidean coordinates from coefficients of $e_1$, $e_2$, and $e_3$ basis vectors in conformal null vector for points
- Example: $(1, 2, 3, 4, 29)$ maps to the point $(2, 3, 4)$
- Derive Euclidean equation for lines and planes from coefficients of basis bivectors in conformal representation
- Example: Bivector $e_1 \wedge e_2 + e_2 \wedge e_3$ represents the plane $x + y = 0$
Role of the Point at Infinity
- Represented by the null vector $(0, 0, 1)$
- Plays a crucial role in representing orientation and direction of lines and planes in Euclidean space
- Lines in opposite directions intersect at the point at infinity, allowing unified treatment of parallel and intersecting lines
- Example: $(1, 1, 0, 0, 1) \wedge (0, 0, 1)$ and $(1, -1, 0, 0, 1) \wedge (0, 0, 1)$ represent parallel lines
- Planes with opposite orientations intersect at the point at infinity, allowing unified treatment of parallel and intersecting planes
- Example: $(1, 0, 0, 0, 0) \wedge (0, 0, 1)$ and $(1, 0, 0, 1, 1) \wedge (0, 0, 1)$ represent parallel planes
Geometric Algebra Operations
- Inner product has natural interpretation corresponding to geometric relationships in Euclidean space
- Angle between vectors
- Distance between points
- Intersection of objects
- Outer product has natural interpretation corresponding to geometric transformations in Euclidean space
- Span of vectors to form higher-grade objects (lines, planes, spheres)
- Rotations and reflections
- Example: $a \wedge b$ represents the oriented plane spanned by vectors $a$ and $b$, and can be used to rotate or reflect other objects